LMFDB - Embedded newform 3150.3.c.f.449.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3150,3,Mod(449,3150)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3150, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 0]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("3150.449");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	3150.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$85.8312832735$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	16.0.9671731157401600000000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 7x^{12} + 753x^{8} + 112x^{4} + 256 $ x^16 + 7x^12 + 753x^8 + 112*x^4 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{28} $
Twist minimal:	no (minimal twist has level 630)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.13
Root			$0.796626 - 0.359610i$ of defining polynomial
Character	$\chi$	$=$	3150.449
Dual form			3150.3.c.f.449.12

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421 q^{2} +2.00000 q^{4} +2.64575i q^{7} +2.82843 q^{8} +O(q^{10})$	$q+1.41421 q^{2} +2.00000 q^{4} +2.64575i q^{7} +2.82843 q^{8} -14.1001i q^{11} -1.78360i q^{13} +3.74166i q^{14} +4.00000 q^{16} -32.2729 q^{17} -9.89548 q^{19} -19.9406i q^{22} -9.58126 q^{23} -2.52238i q^{26} +5.29150i q^{28} +40.5881i q^{29} +48.5626 q^{31} +5.65685 q^{32} -45.6408 q^{34} +62.8976i q^{37} -13.9943 q^{38} +65.7349i q^{41} -50.8420i q^{43} -28.2003i q^{44} -13.5500 q^{46} +65.3405 q^{47} -7.00000 q^{49} -3.56719i q^{52} -64.9575 q^{53} +7.48331i q^{56} +57.4003i q^{58} +111.642i q^{59} +2.69105 q^{61} +68.6778 q^{62} +8.00000 q^{64} +72.1824i q^{67} -64.5458 q^{68} +40.6525i q^{71} +63.6965i q^{73} +88.9506i q^{74} -19.7910 q^{76} +37.3055 q^{77} +82.3035 q^{79} +92.9631i q^{82} +99.9021 q^{83} -71.9014i q^{86} -39.8812i q^{88} -88.5703i q^{89} +4.71895 q^{91} -19.1625 q^{92} +92.4054 q^{94} +117.848i q^{97} -9.89949 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 32 q^{4}+O(q^{10}) $ 16 * q + 32 * q^4	$ 16 q + 32 q^{4} + 64 q^{16} + 160 q^{19} + 224 q^{31} - 192 q^{34} - 64 q^{46} - 112 q^{49} - 288 q^{61} + 128 q^{64} + 320 q^{76} + 128 q^{79} - 448 q^{91}+O(q^{100}) $ 16 * q + 32 * q^4 + 64 * q^16 + 160 * q^19 + 224 * q^31 - 192 * q^34 - 64 * q^46 - 112 * q^49 - 288 * q^61 + 128 * q^64 + 320 * q^76 + 128 * q^79 - 448 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3150\mathbb{Z}\right)^\times$.

$n$	$127$	$451$	$2801$
$\chi(n)$	$-1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.41421	0.707107
$2$	1.41421	0.707107
$3$	0	0
$3$	0	0
$4$	2.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.64575i	0.377964i
$7$	2.64575i	0.377964i
$8$	2.82843	0.353553
$9$	0	0
$10$	0	0
$11$	− 14.1001i	− 1.28183i	−0.767611	−	0.640916i	$-0.778555\pi$
$11$	− 14.1001i	− 1.28183i	0.767611	−	0.640916i	$-0.221445\pi$
$12$	0	0
$13$	− 1.78360i	− 0.137200i	−0.997644	−	0.0685998i	$-0.978147\pi$
$13$	− 1.78360i	− 0.137200i	0.997644	−	0.0685998i	$-0.0218532\pi$
$14$	3.74166i	0.267261i
$15$	0	0
$16$	4.00000	0.250000
$17$	−32.2729	−1.89841	−0.949203	−	0.314665i	$-0.898108\pi$
$17$	−32.2729	−1.89841	−0.949203	+	0.314665i	$0.898108\pi$
$18$	0	0
$19$	−9.89548	−0.520815	−0.260407	−	0.965499i	$-0.583857\pi$
$19$	−9.89548	−0.520815	−0.260407	+	0.965499i	$0.583857\pi$
$20$	0	0
$21$	0	0
$22$	− 19.9406i	− 0.906392i
$23$	−9.58126	−0.416577	−0.208288	−	0.978067i	$-0.566789\pi$
$23$	−9.58126	−0.416577	−0.208288	+	0.978067i	$0.566789\pi$
$24$	0	0
$25$	0	0
$26$	− 2.52238i	− 0.0970148i
$27$	0	0
$28$	5.29150i	0.188982i
$29$	40.5881i	1.39959i	0.714343	+	0.699795i	$0.246726\pi$
$29$	40.5881i	1.39959i	−0.714343	+	0.699795i	$0.753274\pi$
$30$	0	0
$31$	48.5626	1.56653	0.783267	−	0.621685i	$-0.213552\pi$
$31$	48.5626	1.56653	0.783267	+	0.621685i	$0.213552\pi$
$32$	5.65685	0.176777
$33$	0	0
$34$	−45.6408	−1.34238
$35$	0	0
$36$	0	0
$37$	62.8976i	1.69993i	0.526835	+	0.849967i	$0.323378\pi$
$37$	62.8976i	1.69993i	−0.526835	+	0.849967i	$0.676622\pi$
$38$	−13.9943	−0.368272
$39$	0	0
$40$	0	0
$41$	65.7349i	1.60329i	0.597801	+	0.801645i	$0.296041\pi$
$41$	65.7349i	1.60329i	−0.597801	+	0.801645i	$0.703959\pi$
$42$	0	0
$43$	− 50.8420i	− 1.18237i	−0.806535	−	0.591186i	$-0.798660\pi$
$43$	− 50.8420i	− 1.18237i	0.806535	−	0.591186i	$-0.201340\pi$
$44$	− 28.2003i	− 0.640916i
$45$	0	0
$46$	−13.5500	−0.294564
$47$	65.3405	1.39022	0.695111	−	0.718902i	$-0.255355\pi$
$47$	65.3405	1.39022	0.695111	+	0.718902i	$0.255355\pi$
$48$	0	0
$49$	−7.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	− 3.56719i	− 0.0685998i
$53$	−64.9575	−1.22561	−0.612807	−	0.790233i	$-0.709960\pi$
$53$	−64.9575	−1.22561	−0.612807	+	0.790233i	$0.709960\pi$
$54$	0	0
$55$	0	0
$56$	7.48331i	0.133631i
$57$	0	0
$58$	57.4003i	0.989660i
$59$	111.642i	1.89223i	0.323824	+	0.946117i	$0.395032\pi$
$59$	111.642i	1.89223i	−0.323824	+	0.946117i	$0.604968\pi$
$60$	0	0
$61$	2.69105	0.0441156	0.0220578	−	0.999757i	$-0.492978\pi$
$61$	2.69105	0.0441156	0.0220578	+	0.999757i	$0.492978\pi$
$62$	68.6778	1.10771
$63$	0	0
$64$	8.00000	0.125000
$65$	0	0
$66$	0	0
$67$	72.1824i	1.07735i	0.842514	+	0.538674i	$0.181075\pi$
$67$	72.1824i	1.07735i	−0.842514	+	0.538674i	$0.818925\pi$
$68$	−64.5458	−0.949203
$69$	0	0
$70$	0	0
$71$	40.6525i	0.572570i	0.958144	+	0.286285i	$0.0924205\pi$
$71$	40.6525i	0.572570i	−0.958144	+	0.286285i	$0.907580\pi$
$72$	0	0
$73$	63.6965i	0.872554i	0.899812	+	0.436277i	$0.143703\pi$
$73$	63.6965i	0.872554i	−0.899812	+	0.436277i	$0.856297\pi$
$74$	88.9506i	1.20204i
$75$	0	0
$76$	−19.7910	−0.260407
$77$	37.3055	0.484487
$78$	0	0
$79$	82.3035	1.04182	0.520908	−	0.853613i	$-0.325593\pi$
$79$	82.3035	1.04182	0.520908	+	0.853613i	$0.325593\pi$
$80$	0	0
$81$	0	0
$82$	92.9631i	1.13370i
$83$	99.9021	1.20364	0.601820	−	0.798632i	$-0.294442\pi$
$83$	99.9021	1.20364	0.601820	+	0.798632i	$0.294442\pi$
$84$	0	0
$85$	0	0
$86$	− 71.9014i	− 0.836063i
$87$	0	0
$88$	− 39.8812i	− 0.453196i
$89$	− 88.5703i	− 0.995172i	−0.867415	−	0.497586i	$-0.834220\pi$
$89$	− 88.5703i	− 0.995172i	0.867415	−	0.497586i	$-0.165780\pi$
$90$	0	0
$91$	4.71895	0.0518566
$92$	−19.1625	−0.208288
$93$	0	0
$94$	92.4054	0.983036
$95$	0	0
$96$	0	0
$97$	117.848i	1.21493i	0.794348	+	0.607464i	$0.207813\pi$
$97$	117.848i	1.21493i	−0.794348	+	0.607464i	$0.792187\pi$
$98$	−9.89949	−0.101015
$99$	0	0
$100$	0	0
$101$	− 36.0289i	− 0.356722i	−0.983965	−	0.178361i	$-0.942921\pi$
$101$	− 36.0289i	− 0.356722i	0.983965	−	0.178361i	$-0.0570794\pi$
$102$	0	0
$103$	− 34.8611i	− 0.338457i	−0.985577	−	0.169229i	$-0.945872\pi$
$103$	− 34.8611i	− 0.338457i	0.985577	−	0.169229i	$-0.0541276\pi$
$104$	− 5.04477i	− 0.0485074i
$105$	0	0
$106$	−91.8638	−0.866639
$107$	−81.8068	−0.764550	−0.382275	−	0.924049i	$-0.624859\pi$
$107$	−81.8068	−0.764550	−0.382275	+	0.924049i	$0.624859\pi$
$108$	0	0
$109$	−103.430	−0.948903	−0.474452	−	0.880282i	$-0.657354\pi$
$109$	−103.430	−0.948903	−0.474452	+	0.880282i	$0.657354\pi$
$110$	0	0
$111$	0	0
$112$	10.5830i	0.0944911i
$113$	−187.400	−1.65841	−0.829204	−	0.558946i	$-0.811206\pi$
$113$	−187.400	−1.65841	−0.829204	+	0.558946i	$0.811206\pi$
$114$	0	0
$115$	0	0
$116$	81.1763i	0.699795i
$117$	0	0
$118$	157.885i	1.33801i
$119$	− 85.3860i	− 0.717530i
$120$	0	0
$121$	−77.8140	−0.643091
$122$	3.80572	0.0311944
$123$	0	0
$124$	97.1251	0.783267
$125$	0	0
$126$	0	0
$127$	− 81.3040i	− 0.640189i	−0.947386	−	0.320094i	$-0.896285\pi$
$127$	− 81.3040i	− 0.640189i	0.947386	−	0.320094i	$-0.103715\pi$
$128$	11.3137	0.0883883
$129$	0	0
$130$	0	0
$131$	− 9.22925i	− 0.0704523i	−0.999379	−	0.0352261i	$-0.988785\pi$
$131$	− 9.22925i	− 0.0704523i	0.999379	−	0.0352261i	$-0.0112152\pi$
$132$	0	0
$133$	− 26.1810i	− 0.196849i
$134$	102.081i	0.761801i
$135$	0	0
$136$	−91.2815	−0.671188
$137$	27.8171	0.203044	0.101522	−	0.994833i	$-0.467629\pi$
$137$	27.8171	0.203044	0.101522	+	0.994833i	$0.467629\pi$
$138$	0	0
$139$	101.074	0.727147	0.363574	−	0.931565i	$-0.381556\pi$
$139$	101.074	0.727147	0.363574	+	0.931565i	$0.381556\pi$
$140$	0	0
$141$	0	0
$142$	57.4913i	0.404868i
$143$	−25.1489	−0.175867
$144$	0	0
$145$	0	0
$146$	90.0804i	0.616989i
$147$	0	0
$148$	125.795i	0.849967i
$149$	− 119.738i	− 0.803612i	−0.915725	−	0.401806i	$-0.868383\pi$
$149$	− 119.738i	− 0.803612i	0.915725	−	0.401806i	$-0.131617\pi$
$150$	0	0
$151$	41.9026	0.277501	0.138750	−	0.990327i	$-0.455691\pi$
$151$	41.9026	0.277501	0.138750	+	0.990327i	$0.455691\pi$
$152$	−27.9886	−0.184136
$153$	0	0
$154$	52.7579	0.342584
$155$	0	0
$156$	0	0
$157$	130.984i	0.834295i	0.908839	+	0.417148i	$0.136970\pi$
$157$	130.984i	0.834295i	−0.908839	+	0.417148i	$0.863030\pi$
$158$	116.395	0.736676
$159$	0	0
$160$	0	0
$161$	− 25.3496i	− 0.157451i
$162$	0	0
$163$	− 117.726i	− 0.722246i	−0.932518	−	0.361123i	$-0.882393\pi$
$163$	− 117.726i	− 0.722246i	0.932518	−	0.361123i	$-0.117607\pi$
$164$	131.470i	0.801645i
$165$	0	0
$166$	141.283	0.851102
$167$	−208.939	−1.25113	−0.625567	−	0.780170i	$-0.715132\pi$
$167$	−208.939	−1.25113	−0.625567	+	0.780170i	$0.715132\pi$
$168$	0	0
$169$	165.819	0.981176
$170$	0	0
$171$	0	0
$172$	− 101.684i	− 0.591186i
$173$	65.6170	0.379289	0.189644	−	0.981853i	$-0.439266\pi$
$173$	65.6170	0.379289	0.189644	+	0.981853i	$0.439266\pi$
$174$	0	0
$175$	0	0
$176$	− 56.4006i	− 0.320458i
$177$	0	0
$178$	− 125.257i	− 0.703693i
$179$	220.904i	1.23410i	0.786924	+	0.617050i	$0.211672\pi$
$179$	220.904i	1.23410i	−0.786924	+	0.617050i	$0.788328\pi$
$180$	0	0
$181$	−84.5556	−0.467158	−0.233579	−	0.972338i	$-0.575044\pi$
$181$	−84.5556	−0.467158	−0.233579	+	0.972338i	$0.575044\pi$
$182$	6.67360	0.0366681
$183$	0	0
$184$	−27.0999	−0.147282
$185$	0	0
$186$	0	0
$187$	455.052i	2.43344i
$188$	130.681	0.695111
$189$	0	0
$190$	0	0
$191$	308.870i	1.61712i	0.588415	+	0.808559i	$0.299752\pi$
$191$	308.870i	1.61712i	−0.588415	+	0.808559i	$0.700248\pi$
$192$	0	0
$193$	314.793i	1.63105i	0.578720	+	0.815526i	$0.303552\pi$
$193$	314.793i	1.63105i	−0.578720	+	0.815526i	$0.696448\pi$
$194$	166.662i	0.859083i
$195$	0	0
$196$	−14.0000	−0.0714286
$197$	−317.747	−1.61293	−0.806465	−	0.591282i	$-0.798622\pi$
$197$	−317.747	−1.61293	−0.806465	+	0.591282i	$0.798622\pi$
$198$	0	0
$199$	−123.082	−0.618502	−0.309251	−	0.950980i	$-0.600078\pi$
$199$	−123.082	−0.618502	−0.309251	+	0.950980i	$0.600078\pi$
$200$	0	0
$201$	0	0
$202$	− 50.9526i	− 0.252240i
$203$	−107.386	−0.528996
$204$	0	0
$205$	0	0
$206$	− 49.3010i	− 0.239325i
$207$	0	0
$208$	− 7.13438i	− 0.0342999i
$209$	139.528i	0.667596i
$210$	0	0
$211$	−298.146	−1.41302	−0.706508	−	0.707705i	$-0.749730\pi$
$211$	−298.146	−1.41302	−0.706508	+	0.707705i	$0.749730\pi$
$212$	−129.915	−0.612807
$213$	0	0
$214$	−115.692	−0.540618
$215$	0	0
$216$	0	0
$217$	128.484i	0.592094i
$218$	−146.273	−0.670976
$219$	0	0
$220$	0	0
$221$	57.5618i	0.260460i
$222$	0	0
$223$	− 83.8687i	− 0.376093i	−0.982160	−	0.188046i	$-0.939784\pi$
$223$	− 83.8687i	− 0.376093i	0.982160	−	0.188046i	$-0.0602155\pi$
$224$	14.9666i	0.0668153i
$225$	0	0
$226$	−265.024	−1.17267
$227$	94.6100	0.416784	0.208392	−	0.978045i	$-0.433177\pi$
$227$	94.6100	0.416784	0.208392	+	0.978045i	$0.433177\pi$
$228$	0	0
$229$	318.779	1.39205	0.696025	−	0.718018i	$-0.254950\pi$
$229$	318.779	1.39205	0.696025	+	0.718018i	$0.254950\pi$
$230$	0	0
$231$	0	0
$232$	114.801i	0.494830i
$233$	187.832	0.806147	0.403073	−	0.915168i	$-0.367942\pi$
$233$	187.832	0.806147	0.403073	+	0.915168i	$0.367942\pi$
$234$	0	0
$235$	0	0
$236$	223.284i	0.946117i
$237$	0	0
$238$	− 120.754i	− 0.507370i
$239$	− 361.090i	− 1.51084i	−0.655243	−	0.755418i	$-0.727434\pi$
$239$	− 361.090i	− 1.51084i	0.655243	−	0.755418i	$-0.272566\pi$
$240$	0	0
$241$	−14.5899	−0.0605391	−0.0302695	−	0.999542i	$-0.509637\pi$
$241$	−14.5899	−0.0605391	−0.0302695	+	0.999542i	$0.509637\pi$
$242$	−110.046	−0.454734
$243$	0	0
$244$	5.38210	0.0220578
$245$	0	0
$246$	0	0
$247$	17.6495i	0.0714556i
$248$	137.356	0.553854
$249$	0	0
$250$	0	0
$251$	359.495i	1.43225i	0.697971	+	0.716126i	$0.254086\pi$
$251$	359.495i	1.43225i	−0.697971	+	0.716126i	$0.745914\pi$
$252$	0	0
$253$	135.097i	0.533981i
$254$	− 114.981i	− 0.452682i
$255$	0	0
$256$	16.0000	0.0625000
$257$	196.720	0.765447	0.382724	−	0.923863i	$-0.374986\pi$
$257$	196.720	0.765447	0.382724	+	0.923863i	$0.374986\pi$
$258$	0	0
$259$	−166.411	−0.642515
$260$	0	0
$261$	0	0
$262$	− 13.0521i	− 0.0498173i
$263$	−396.028	−1.50581	−0.752905	−	0.658129i	$-0.771348\pi$
$263$	−396.028	−1.50581	−0.752905	+	0.658129i	$0.771348\pi$
$264$	0	0
$265$	0	0
$266$	− 37.0255i	− 0.139194i
$267$	0	0
$268$	144.365i	0.538674i
$269$	203.497i	0.756494i	0.925705	+	0.378247i	$0.123473\pi$
$269$	203.497i	0.756494i	−0.925705	+	0.378247i	$0.876527\pi$
$270$	0	0
$271$	−173.701	−0.640964	−0.320482	−	0.947255i	$-0.603845\pi$
$271$	−173.701	−0.640964	−0.320482	+	0.947255i	$0.603845\pi$
$272$	−129.092	−0.474601
$273$	0	0
$274$	39.3393	0.143574
$275$	0	0
$276$	0	0
$277$	− 130.018i	− 0.469380i	−0.972070	−	0.234690i	$-0.924593\pi$
$277$	− 130.018i	− 0.469380i	0.972070	−	0.234690i	$-0.0754075\pi$
$278$	142.940	0.514171
$279$	0	0
$280$	0	0
$281$	− 476.464i	− 1.69560i	−0.530314	−	0.847801i	$-0.677926\pi$
$281$	− 476.464i	− 1.69560i	0.530314	−	0.847801i	$-0.322074\pi$
$282$	0	0
$283$	56.5824i	0.199938i	0.994991	+	0.0999690i	$0.0318744\pi$
$283$	56.5824i	0.199938i	−0.994991	+	0.0999690i	$0.968126\pi$
$284$	81.3050i	0.286285i
$285$	0	0
$286$	−35.5660	−0.124357
$287$	−173.918	−0.605986
$288$	0	0
$289$	752.540	2.60394
$290$	0	0
$291$	0	0
$292$	127.393i	0.436277i
$293$	124.651	0.425431	0.212715	−	0.977114i	$-0.431769\pi$
$293$	124.651	0.425431	0.212715	+	0.977114i	$0.431769\pi$
$294$	0	0
$295$	0	0
$296$	177.901i	0.601018i
$297$	0	0
$298$	− 169.335i	− 0.568239i
$299$	17.0891i	0.0571542i
$300$	0	0
$301$	134.515	0.446895
$302$	59.2592	0.196223
$303$	0	0
$304$	−39.5819	−0.130204
$305$	0	0
$306$	0	0
$307$	27.2322i	0.0887042i	0.999016	+	0.0443521i	$0.0141223\pi$
$307$	27.2322i	0.0887042i	−0.999016	+	0.0443521i	$0.985878\pi$
$308$	74.6109	0.242243
$309$	0	0
$310$	0	0
$311$	− 63.9207i	− 0.205533i	−0.994706	−	0.102766i	$-0.967231\pi$
$311$	− 63.9207i	− 0.205533i	0.994706	−	0.102766i	$-0.0327694\pi$
$312$	0	0
$313$	− 277.105i	− 0.885320i	−0.896689	−	0.442660i	$-0.854035\pi$
$313$	− 277.105i	− 0.885320i	0.896689	−	0.442660i	$-0.145965\pi$
$314$	185.240i	0.589936i
$315$	0	0
$316$	164.607	0.520908
$317$	−407.861	−1.28663	−0.643313	−	0.765603i	$-0.722441\pi$
$317$	−407.861	−1.28663	−0.643313	+	0.765603i	$0.722441\pi$
$318$	0	0
$319$	572.299	1.79404
$320$	0	0
$321$	0	0
$322$	− 35.8498i	− 0.111335i
$323$	319.356	0.988717
$324$	0	0
$325$	0	0
$326$	− 166.490i	− 0.510705i
$327$	0	0
$328$	185.926i	0.566848i
$329$	172.875i	0.525455i
$330$	0	0
$331$	350.438	1.05872	0.529362	−	0.848396i	$-0.322431\pi$
$331$	350.438	1.05872	0.529362	+	0.848396i	$0.322431\pi$
$332$	199.804	0.601820
$333$	0	0
$334$	−295.485	−0.884686
$335$	0	0
$336$	0	0
$337$	506.953i	1.50431i	0.658985	+	0.752156i	$0.270986\pi$
$337$	506.953i	1.50431i	−0.658985	+	0.752156i	$0.729014\pi$
$338$	234.503	0.693796
$339$	0	0
$340$	0	0
$341$	− 684.739i	− 2.00803i
$342$	0	0
$343$	− 18.5203i	− 0.0539949i
$344$	− 143.803i	− 0.418032i
$345$	0	0
$346$	92.7964	0.268198
$347$	−143.703	−0.414131	−0.207065	−	0.978327i	$-0.566391\pi$
$347$	−143.703	−0.414131	−0.207065	+	0.978327i	$0.566391\pi$
$348$	0	0
$349$	168.383	0.482473	0.241237	−	0.970466i	$-0.422447\pi$
$349$	168.383	0.482473	0.241237	+	0.970466i	$0.422447\pi$
$350$	0	0
$351$	0	0
$352$	− 79.7625i	− 0.226598i
$353$	−531.562	−1.50584	−0.752921	−	0.658111i	$-0.771356\pi$
$353$	−531.562	−1.50584	−0.752921	+	0.658111i	$0.771356\pi$
$354$	0	0
$355$	0	0
$356$	− 177.141i	− 0.497586i
$357$	0	0
$358$	312.405i	0.872640i
$359$	− 63.2884i	− 0.176291i	−0.996108	−	0.0881454i	$-0.971906\pi$
$359$	− 63.2884i	− 0.176291i	0.996108	−	0.0881454i	$-0.0280940\pi$
$360$	0	0
$361$	−263.080	−0.728752
$362$	−119.580	−0.330331
$363$	0	0
$364$	9.43790	0.0259283
$365$	0	0
$366$	0	0
$367$	252.555i	0.688160i	0.938940	+	0.344080i	$0.111809\pi$
$367$	252.555i	0.688160i	−0.938940	+	0.344080i	$0.888191\pi$
$368$	−38.3251	−0.104144
$369$	0	0
$370$	0	0
$371$	− 171.861i	− 0.463238i
$372$	0	0
$373$	223.925i	0.600336i	0.953886	+	0.300168i	$0.0970427\pi$
$373$	223.925i	0.600336i	−0.953886	+	0.300168i	$0.902957\pi$
$374$	643.541i	1.72070i
$375$	0	0
$376$	184.811	0.491518
$377$	72.3928	0.192023
$378$	0	0
$379$	503.313	1.32800	0.664001	−	0.747732i	$-0.268857\pi$
$379$	503.313	1.32800	0.664001	+	0.747732i	$0.268857\pi$
$380$	0	0
$381$	0	0
$382$	436.808i	1.14348i
$383$	−446.558	−1.16595	−0.582974	−	0.812491i	$-0.698111\pi$
$383$	−446.558	−1.16595	−0.582974	+	0.812491i	$0.698111\pi$
$384$	0	0
$385$	0	0
$386$	445.185i	1.15333i
$387$	0	0
$388$	235.696i	0.607464i
$389$	18.4450i	0.0474165i	0.999719	+	0.0237083i	$0.00754728\pi$
$389$	18.4450i	0.0474165i	−0.999719	+	0.0237083i	$0.992453\pi$
$390$	0	0
$391$	309.215	0.790831
$392$	−19.7990	−0.0505076
$393$	0	0
$394$	−449.362	−1.14051
$395$	0	0
$396$	0	0
$397$	− 209.590i	− 0.527934i	−0.964532	−	0.263967i	$-0.914969\pi$
$397$	− 209.590i	− 0.527934i	0.964532	−	0.263967i	$-0.0850310\pi$
$398$	−174.064	−0.437347
$399$	0	0
$400$	0	0
$401$	− 277.998i	− 0.693261i	−0.938002	−	0.346631i	$-0.887326\pi$
$401$	− 277.998i	− 0.693261i	0.938002	−	0.346631i	$-0.112674\pi$
$402$	0	0
$403$	− 86.6160i	− 0.214928i
$404$	− 72.0578i	− 0.178361i
$405$	0	0
$406$	−151.867	−0.374056
$407$	886.865	2.17903
$408$	0	0
$409$	506.079	1.23736	0.618678	−	0.785644i	$-0.287668\pi$
$409$	506.079	1.23736	0.618678	+	0.785644i	$0.287668\pi$
$410$	0	0
$411$	0	0
$412$	− 69.7221i	− 0.169229i
$413$	−295.377	−0.715198
$414$	0	0
$415$	0	0
$416$	− 10.0895i	− 0.0242537i
$417$	0	0
$418$	197.322i	0.472062i
$419$	− 95.9760i	− 0.229060i	−0.993420	−	0.114530i	$-0.963464\pi$
$419$	− 95.9760i	− 0.229060i	0.993420	−	0.114530i	$-0.0365362\pi$
$420$	0	0
$421$	−236.741	−0.562330	−0.281165	−	0.959659i	$-0.590721\pi$
$421$	−236.741	−0.562330	−0.281165	+	0.959659i	$0.590721\pi$
$422$	−421.642	−0.999153
$423$	0	0
$424$	−183.728	−0.433320
$425$	0	0
$426$	0	0
$427$	7.11985i	0.0166741i
$428$	−163.614	−0.382275
$429$	0	0
$430$	0	0
$431$	− 494.406i	− 1.14711i	−0.819166	−	0.573557i	$-0.805563\pi$
$431$	− 494.406i	− 1.14711i	0.819166	−	0.573557i	$-0.194437\pi$
$432$	0	0
$433$	426.179i	0.984247i	0.870526	+	0.492123i	$0.163779\pi$
$433$	426.179i	0.984247i	−0.870526	+	0.492123i	$0.836221\pi$
$434$	181.705i	0.418674i
$435$	0	0
$436$	−206.861	−0.474452
$437$	94.8112	0.216959
$438$	0	0
$439$	−374.522	−0.853125	−0.426562	−	0.904458i	$-0.640276\pi$
$439$	−374.522	−0.853125	−0.426562	+	0.904458i	$0.640276\pi$
$440$	0	0
$441$	0	0
$442$	81.4046i	0.184173i
$443$	38.1307	0.0860739	0.0430370	−	0.999073i	$-0.486297\pi$
$443$	38.1307	0.0860739	0.0430370	+	0.999073i	$0.486297\pi$
$444$	0	0
$445$	0	0
$446$	− 118.608i	− 0.265938i
$447$	0	0
$448$	21.1660i	0.0472456i
$449$	1.52774i	0.00340253i	0.999999	+	0.00170126i	$0.000541529\pi$
$449$	1.52774i	0.00340253i	−0.999999	+	0.00170126i	$0.999458\pi$
$450$	0	0
$451$	926.871	2.05515
$452$	−374.800	−0.829204
$453$	0	0
$454$	133.799	0.294711
$455$	0	0
$456$	0	0
$457$	− 141.016i	− 0.308569i	−0.988026	−	0.154284i	$-0.950693\pi$
$457$	− 141.016i	− 0.308569i	0.988026	−	0.154284i	$-0.0493072\pi$
$458$	450.822	0.984327
$459$	0	0
$460$	0	0
$461$	10.4463i	0.0226601i	0.999936	+	0.0113301i	$0.00360655\pi$
$461$	10.4463i	0.0226601i	−0.999936	+	0.0113301i	$0.996393\pi$
$462$	0	0
$463$	− 159.805i	− 0.345152i	−0.984996	−	0.172576i	$-0.944791\pi$
$463$	− 159.805i	− 0.345152i	0.984996	−	0.172576i	$-0.0552091\pi$
$464$	162.353i	0.349898i
$465$	0	0
$466$	265.635	0.570032
$467$	58.0764	0.124361	0.0621803	−	0.998065i	$-0.480195\pi$
$467$	58.0764	0.124361	0.0621803	+	0.998065i	$0.480195\pi$
$468$	0	0
$469$	−190.977	−0.407200
$470$	0	0
$471$	0	0
$472$	315.771i	0.669006i
$473$	−716.879	−1.51560
$474$	0	0
$475$	0	0
$476$	− 170.772i	− 0.358765i
$477$	0	0
$478$	− 510.658i	− 1.06832i
$479$	− 134.072i	− 0.279899i	−0.990159	−	0.139950i	$-0.955306\pi$
$479$	− 134.072i	− 0.279899i	0.990159	−	0.139950i	$-0.0446940\pi$
$480$	0	0
$481$	112.184	0.233230
$482$	−20.6333	−0.0428076
$483$	0	0
$484$	−155.628	−0.321546
$485$	0	0
$486$	0	0
$487$	277.859i	0.570552i	0.958445	+	0.285276i	$0.0920853\pi$
$487$	277.859i	0.570552i	−0.958445	+	0.285276i	$0.907915\pi$
$488$	7.61144	0.0155972
$489$	0	0
$490$	0	0
$491$	514.975i	1.04883i	0.851463	+	0.524414i	$0.175716\pi$
$491$	514.975i	1.04883i	−0.851463	+	0.524414i	$0.824284\pi$
$492$	0	0
$493$	− 1309.90i	− 2.65699i
$494$	24.9602i	0.0505267i
$495$	0	0
$496$	194.250	0.391634
$497$	−107.556	−0.216411
$498$	0	0
$499$	144.052	0.288682	0.144341	−	0.989528i	$-0.453894\pi$
$499$	144.052	0.288682	0.144341	+	0.989528i	$0.453894\pi$
$500$	0	0
$501$	0	0
$502$	508.403i	1.01275i
$503$	−596.609	−1.18610	−0.593051	−	0.805165i	$-0.702077\pi$
$503$	−596.609	−1.18610	−0.593051	+	0.805165i	$0.702077\pi$
$504$	0	0
$505$	0	0
$506$	191.056i	0.377582i
$507$	0	0
$508$	− 162.608i	− 0.320094i
$509$	7.91201i	0.0155442i	0.999970	+	0.00777211i	$0.00247397\pi$
$509$	7.91201i	0.0155442i	−0.999970	+	0.00777211i	$0.997526\pi$
$510$	0	0
$511$	−168.525	−0.329794
$512$	22.6274	0.0441942
$513$	0	0
$514$	278.204	0.541253
$515$	0	0
$516$	0	0
$517$	− 921.310i	− 1.78203i
$518$	−235.341	−0.454327
$519$	0	0
$520$	0	0
$521$	− 746.931i	− 1.43365i	−0.697253	−	0.716825i	$-0.745595\pi$
$521$	− 746.931i	− 1.43365i	0.697253	−	0.716825i	$-0.254405\pi$
$522$	0	0
$523$	138.896i	0.265576i	0.991144	+	0.132788i	$0.0423929\pi$
$523$	138.896i	0.265576i	−0.991144	+	0.132788i	$0.957607\pi$
$524$	− 18.4585i	− 0.0352261i
$525$	0	0
$526$	−560.068	−1.06477
$527$	−1567.25	−2.97392
$528$	0	0
$529$	−437.199	−0.826464
$530$	0	0
$531$	0	0
$532$	− 52.3619i	− 0.0984247i
$533$	117.244	0.219971
$534$	0	0
$535$	0	0
$536$	204.163i	0.380900i
$537$	0	0
$538$	287.788i	0.534922i
$539$	98.7010i	0.183119i
$540$	0	0
$541$	−684.033	−1.26439	−0.632193	−	0.774811i	$-0.717845\pi$
$541$	−684.033	−1.26439	−0.632193	+	0.774811i	$0.717845\pi$
$542$	−245.651	−0.453230
$543$	0	0
$544$	−182.563	−0.335594
$545$	0	0
$546$	0	0
$547$	485.135i	0.886901i	0.896299	+	0.443451i	$0.146246\pi$
$547$	485.135i	0.886901i	−0.896299	+	0.443451i	$0.853754\pi$
$548$	55.6341	0.101522
$549$	0	0
$550$	0	0
$551$	− 401.639i	− 0.728927i
$552$	0	0
$553$	217.755i	0.393770i
$554$	− 183.874i	− 0.331902i
$555$	0	0
$556$	202.147	0.363574
$557$	227.641	0.408692	0.204346	−	0.978899i	$-0.434493\pi$
$557$	227.641	0.408692	0.204346	+	0.978899i	$0.434493\pi$
$558$	0	0
$559$	−90.6815	−0.162221
$560$	0	0
$561$	0	0
$562$	− 673.822i	− 1.19897i
$563$	69.7030	0.123806	0.0619032	−	0.998082i	$-0.480283\pi$
$563$	69.7030	0.123806	0.0619032	+	0.998082i	$0.480283\pi$
$564$	0	0
$565$	0	0
$566$	80.0197i	0.141378i
$567$	0	0
$568$	114.983i	0.202434i
$569$	573.975i	1.00874i	0.863487	+	0.504372i	$0.168276\pi$
$569$	573.975i	1.00874i	−0.863487	+	0.504372i	$0.831724\pi$
$570$	0	0
$571$	289.675	0.507312	0.253656	−	0.967294i	$-0.418367\pi$
$571$	289.675	0.507312	0.253656	+	0.967294i	$0.418367\pi$
$572$	−50.2979	−0.0879334
$573$	0	0
$574$	−245.957	−0.428497
$575$	0	0
$576$	0	0
$577$	122.652i	0.212569i	0.994336	+	0.106285i	$0.0338955\pi$
$577$	122.652i	0.212569i	−0.994336	+	0.106285i	$0.966105\pi$
$578$	1064.25	1.84127
$579$	0	0
$580$	0	0
$581$	264.316i	0.454933i
$582$	0	0
$583$	915.910i	1.57103i
$584$	180.161i	0.308494i
$585$	0	0
$586$	176.283	0.300825
$587$	−156.694	−0.266940	−0.133470	−	0.991053i	$-0.542612\pi$
$587$	−156.694	−0.266940	−0.133470	+	0.991053i	$0.542612\pi$
$588$	0	0
$589$	−480.550	−0.815874
$590$	0	0
$591$	0	0
$592$	251.590i	0.424984i
$593$	78.4823	0.132348	0.0661739	−	0.997808i	$-0.478921\pi$
$593$	78.4823	0.132348	0.0661739	+	0.997808i	$0.478921\pi$
$594$	0	0
$595$	0	0
$596$	− 239.476i	− 0.401806i
$597$	0	0
$598$	24.1676i	0.0404141i
$599$	− 104.010i	− 0.173639i	−0.996224	−	0.0868195i	$-0.972330\pi$
$599$	− 104.010i	− 0.173639i	0.996224	−	0.0868195i	$-0.0276703\pi$
$600$	0	0
$601$	773.253	1.28661	0.643305	−	0.765610i	$-0.277563\pi$
$601$	773.253	1.28661	0.643305	+	0.765610i	$0.277563\pi$
$602$	190.233	0.316002
$603$	0	0
$604$	83.8052	0.138750
$605$	0	0
$606$	0	0
$607$	349.051i	0.575044i	0.957774	+	0.287522i	$0.0928314\pi$
$607$	349.051i	0.575044i	−0.957774	+	0.287522i	$0.907169\pi$
$608$	−55.9773	−0.0920679
$609$	0	0
$610$	0	0
$611$	− 116.541i	− 0.190738i
$612$	0	0
$613$	323.779i	0.528188i	0.964497	+	0.264094i	$0.0850729\pi$
$613$	323.779i	0.528188i	−0.964497	+	0.264094i	$0.914927\pi$
$614$	38.5121i	0.0627233i
$615$	0	0
$616$	105.516	0.171292
$617$	−1040.96	−1.68713	−0.843567	−	0.537023i	$-0.819549\pi$
$617$	−1040.96	−1.68713	−0.843567	+	0.537023i	$0.819549\pi$
$618$	0	0
$619$	783.600	1.26591	0.632957	−	0.774187i	$-0.281841\pi$
$619$	783.600	1.26591	0.632957	+	0.774187i	$0.281841\pi$
$620$	0	0
$621$	0	0
$622$	− 90.3976i	− 0.145334i
$623$	234.335	0.376140
$624$	0	0
$625$	0	0
$626$	− 391.886i	− 0.626016i
$627$	0	0
$628$	261.969i	0.417148i
$629$	− 2029.89i	− 3.22717i
$630$	0	0
$631$	967.921	1.53395	0.766974	−	0.641678i	$-0.221762\pi$
$631$	967.921	1.53395	0.766974	+	0.641678i	$0.221762\pi$
$632$	232.790	0.368338
$633$	0	0
$634$	−576.802	−0.909783
$635$	0	0
$636$	0	0
$637$	12.4852i	0.0195999i
$638$	809.352	1.26858
$639$	0	0
$640$	0	0
$641$	38.9129i	0.0607066i	0.999539	+	0.0303533i	$0.00966324\pi$
$641$	38.9129i	0.0607066i	−0.999539	+	0.0303533i	$0.990337\pi$
$642$	0	0
$643$	− 1123.77i	− 1.74771i	−0.486191	−	0.873853i	$-0.661614\pi$
$643$	− 1123.77i	− 1.74771i	0.486191	−	0.873853i	$-0.338386\pi$
$644$	− 50.6993i	− 0.0787256i
$645$	0	0
$646$	451.637	0.699129
$647$	−635.461	−0.982165	−0.491083	−	0.871113i	$-0.663399\pi$
$647$	−635.461	−0.982165	−0.491083	+	0.871113i	$0.663399\pi$
$648$	0	0
$649$	1574.17	2.42553
$650$	0	0
$651$	0	0
$652$	− 235.452i	− 0.361123i
$653$	361.063	0.552929	0.276465	−	0.961024i	$-0.410837\pi$
$653$	361.063	0.552929	0.276465	+	0.961024i	$0.410837\pi$
$654$	0	0
$655$	0	0
$656$	262.939i	0.400822i
$657$	0	0
$658$	244.482i	0.371553i
$659$	− 1284.71i	− 1.94949i	−0.223327	−	0.974744i	$-0.571692\pi$
$659$	− 1284.71i	− 1.94949i	0.223327	−	0.974744i	$-0.428308\pi$
$660$	0	0
$661$	605.603	0.916192	0.458096	−	0.888903i	$-0.348532\pi$
$661$	605.603	0.916192	0.458096	+	0.888903i	$0.348532\pi$
$662$	495.594	0.748632
$663$	0	0
$664$	282.566	0.425551
$665$	0	0
$666$	0	0
$667$	− 388.886i	− 0.583037i
$668$	−417.879	−0.625567
$669$	0	0
$670$	0	0
$671$	− 37.9442i	− 0.0565487i
$672$	0	0
$673$	863.984i	1.28378i	0.766797	+	0.641890i	$0.221849\pi$
$673$	863.984i	1.28378i	−0.766797	+	0.641890i	$0.778151\pi$
$674$	716.940i	1.06371i
$675$	0	0
$676$	331.638	0.490588
$677$	1009.59	1.49127	0.745635	−	0.666354i	$-0.232146\pi$
$677$	1009.59	1.49127	0.745635	+	0.666354i	$0.232146\pi$
$678$	0	0
$679$	−311.796	−0.459199
$680$	0	0
$681$	0	0
$682$	− 968.367i	− 1.41989i
$683$	−786.887	−1.15210	−0.576052	−	0.817413i	$-0.695407\pi$
$683$	−786.887	−1.15210	−0.576052	+	0.817413i	$0.695407\pi$
$684$	0	0
$685$	0	0
$686$	− 26.1916i	− 0.0381802i
$687$	0	0
$688$	− 203.368i	− 0.295593i
$689$	115.858i	0.168154i
$690$	0	0
$691$	−1285.82	−1.86081	−0.930405	−	0.366534i	$-0.880544\pi$
$691$	−1285.82	−1.86081	−0.930405	+	0.366534i	$0.880544\pi$
$692$	131.234	0.189644
$693$	0	0
$694$	−203.227	−0.292835
$695$	0	0
$696$	0	0
$697$	− 2121.45i	− 3.04369i
$698$	238.130	0.341160
$699$	0	0
$700$	0	0
$701$	− 717.153i	− 1.02304i	−0.859271	−	0.511521i	$-0.829082\pi$
$701$	− 717.153i	− 1.02304i	0.859271	−	0.511521i	$-0.170918\pi$
$702$	0	0
$703$	− 622.402i	− 0.885351i
$704$	− 112.801i	− 0.160229i
$705$	0	0
$706$	−751.742	−1.06479
$707$	95.3235	0.134828
$708$	0	0
$709$	−197.923	−0.279158	−0.139579	−	0.990211i	$-0.544575\pi$
$709$	−197.923	−0.279158	−0.139579	+	0.990211i	$0.544575\pi$
$710$	0	0
$711$	0	0
$712$	− 250.515i	− 0.351847i
$713$	−465.291	−0.652582
$714$	0	0
$715$	0	0
$716$	441.808i	0.617050i
$717$	0	0
$718$	− 89.5033i	− 0.124656i
$719$	168.511i	0.234369i	0.993110	+	0.117185i	$0.0373869\pi$
$719$	168.511i	0.234369i	−0.993110	+	0.117185i	$0.962613\pi$
$720$	0	0
$721$	92.2337	0.127925
$722$	−372.051	−0.515306
$723$	0	0
$724$	−169.111	−0.233579
$725$	0	0
$726$	0	0
$727$	906.873i	1.24742i	0.781657	+	0.623709i	$0.214375\pi$
$727$	906.873i	1.24742i	−0.781657	+	0.623709i	$0.785625\pi$
$728$	13.3472	0.0183341
$729$	0	0
$730$	0	0
$731$	1640.82i	2.24462i
$732$	0	0
$733$	22.6632i	0.0309184i	0.999880	+	0.0154592i	$0.00492101\pi$
$733$	22.6632i	0.0309184i	−0.999880	+	0.0154592i	$0.995079\pi$
$734$	357.166i	0.486602i
$735$	0	0
$736$	−54.1998	−0.0736410
$737$	1017.78	1.38098
$738$	0	0
$739$	−614.258	−0.831202	−0.415601	−	0.909547i	$-0.636429\pi$
$739$	−614.258	−0.831202	−0.415601	+	0.909547i	$0.636429\pi$
$740$	0	0
$741$	0	0
$742$	− 243.049i	− 0.327559i
$743$	−33.0291	−0.0444537	−0.0222268	−	0.999753i	$-0.507076\pi$
$743$	−33.0291	−0.0444537	−0.0222268	+	0.999753i	$0.507076\pi$
$744$	0	0
$745$	0	0
$746$	316.678i	0.424501i
$747$	0	0
$748$	910.105i	1.21672i
$749$	− 216.441i	− 0.288973i
$750$	0	0
$751$	−905.403	−1.20560	−0.602798	−	0.797894i	$-0.705948\pi$
$751$	−905.403	−1.20560	−0.602798	+	0.797894i	$0.705948\pi$
$752$	261.362	0.347556
$753$	0	0
$754$	102.379	0.135781
$755$	0	0
$756$	0	0
$757$	64.4031i	0.0850768i	0.999095	+	0.0425384i	$0.0135445\pi$
$757$	64.4031i	0.0850768i	−0.999095	+	0.0425384i	$0.986456\pi$
$758$	711.791	0.939039
$759$	0	0
$760$	0	0
$761$	508.718i	0.668486i	0.942487	+	0.334243i	$0.108481\pi$
$761$	508.718i	0.668486i	−0.942487	+	0.334243i	$0.891519\pi$
$762$	0	0
$763$	− 273.651i	− 0.358652i
$764$	617.739i	0.808559i
$765$	0	0
$766$	−631.528	−0.824449
$767$	199.124	0.259614
$768$	0	0
$769$	934.507	1.21522	0.607612	−	0.794234i	$-0.292128\pi$
$769$	934.507	1.21522	0.607612	+	0.794234i	$0.292128\pi$
$770$	0	0
$771$	0	0
$772$	629.586i	0.815526i
$773$	1013.90	1.31164	0.655821	−	0.754917i	$-0.272323\pi$
$773$	1013.90	1.31164	0.655821	+	0.754917i	$0.272323\pi$
$774$	0	0
$775$	0	0
$776$	333.324i	0.429542i
$777$	0	0
$778$	26.0852i	0.0335285i
$779$	− 650.478i	− 0.835017i
$780$	0	0
$781$	573.206	0.733939
$782$	437.296	0.559202
$783$	0	0
$784$	−28.0000	−0.0357143
$785$	0	0
$786$	0	0
$787$	374.605i	0.475991i	0.971266	+	0.237996i	$0.0764904\pi$
$787$	374.605i	0.475991i	−0.971266	+	0.237996i	$0.923510\pi$
$788$	−635.494	−0.806465
$789$	0	0
$790$	0	0
$791$	− 495.814i	− 0.626819i
$792$	0	0
$793$	− 4.79975i	− 0.00605264i
$794$	− 296.405i	− 0.373306i
$795$	0	0
$796$	−246.164	−0.309251
$797$	734.515	0.921600	0.460800	−	0.887504i	$-0.347563\pi$
$797$	734.515	0.921600	0.460800	+	0.887504i	$0.347563\pi$
$798$	0	0
$799$	−2108.73	−2.63921
$800$	0	0
$801$	0	0
$802$	− 393.148i	− 0.490210i
$803$	898.129	1.11847
$804$	0	0
$805$	0	0
$806$	− 122.493i	− 0.151977i
$807$	0	0
$808$	− 101.905i	− 0.126120i
$809$	− 212.258i	− 0.262371i	−0.991358	−	0.131186i	$-0.958122\pi$
$809$	− 212.258i	− 0.262371i	0.991358	−	0.131186i	$-0.0418784\pi$
$810$	0	0
$811$	−1343.78	−1.65694	−0.828470	−	0.560034i	$-0.810788\pi$
$811$	−1343.78	−1.65694	−0.828470	+	0.560034i	$0.810788\pi$
$812$	−214.772	−0.264498
$813$	0	0
$814$	1254.22	1.54081
$815$	0	0
$816$	0	0
$817$	503.106i	0.615797i
$818$	715.703	0.874943
$819$	0	0
$820$	0	0
$821$	− 126.090i	− 0.153582i	−0.997047	−	0.0767908i	$-0.975533\pi$
$821$	− 126.090i	− 0.153582i	0.997047	−	0.0767908i	$-0.0244674\pi$
$822$	0	0
$823$	74.0891i	0.0900232i	0.998986	+	0.0450116i	$0.0143325\pi$
$823$	74.0891i	0.0900232i	−0.998986	+	0.0450116i	$0.985668\pi$
$824$	− 98.6020i	− 0.119663i
$825$	0	0
$826$	−417.726	−0.505721
$827$	626.689	0.757786	0.378893	−	0.925441i	$-0.376305\pi$
$827$	626.689	0.757786	0.378893	+	0.925441i	$0.376305\pi$
$828$	0	0
$829$	−605.458	−0.730348	−0.365174	−	0.930939i	$-0.618990\pi$
$829$	−605.458	−0.730348	−0.365174	+	0.930939i	$0.618990\pi$
$830$	0	0
$831$	0	0
$832$	− 14.2688i	− 0.0171500i
$833$	225.910	0.271201
$834$	0	0
$835$	0	0
$836$	279.055i	0.333798i
$837$	0	0
$838$	− 135.731i	− 0.161970i
$839$	− 1303.81i	− 1.55400i	−0.629499	−	0.777001i	$-0.716740\pi$
$839$	− 1303.81i	− 1.55400i	0.629499	−	0.777001i	$-0.283260\pi$
$840$	0	0
$841$	−806.397	−0.958855
$842$	−334.802	−0.397627
$843$	0	0
$844$	−596.293	−0.706508
$845$	0	0
$846$	0	0
$847$	− 205.877i	− 0.243066i
$848$	−259.830	−0.306403
$849$	0	0
$850$	0	0
$851$	− 602.638i	− 0.708153i
$852$	0	0
$853$	34.1253i	0.0400062i	0.999800	+	0.0200031i	$0.00636761\pi$
$853$	34.1253i	0.0400062i	−0.999800	+	0.0200031i	$0.993632\pi$
$854$	10.0690i	0.0117904i
$855$	0	0
$856$	−231.385	−0.270309
$857$	−1513.23	−1.76573	−0.882866	−	0.469626i	$-0.844389\pi$
$857$	−1513.23	−1.76573	−0.882866	+	0.469626i	$0.844389\pi$
$858$	0	0
$859$	1089.68	1.26855	0.634273	−	0.773109i	$-0.281299\pi$
$859$	1089.68	1.26855	0.634273	+	0.773109i	$0.281299\pi$
$860$	0	0
$861$	0	0
$862$	− 699.196i	− 0.811132i
$863$	643.590	0.745759	0.372880	−	0.927880i	$-0.378370\pi$
$863$	643.590	0.745759	0.372880	+	0.927880i	$0.378370\pi$
$864$	0	0
$865$	0	0
$866$	602.708i	0.695967i
$867$	0	0
$868$	256.969i	0.296047i
$869$	− 1160.49i	− 1.33543i
$870$	0	0
$871$	128.744	0.147812
$872$	−292.545	−0.335488
$873$	0	0
$874$	134.083	0.153413
$875$	0	0
$876$	0	0
$877$	− 766.776i	− 0.874317i	−0.899385	−	0.437158i	$-0.855985\pi$
$877$	− 766.776i	− 0.874317i	0.899385	−	0.437158i	$-0.144015\pi$
$878$	−529.654	−0.603250
$879$	0	0
$880$	0	0
$881$	157.211i	0.178446i	0.996012	+	0.0892229i	$0.0284384\pi$
$881$	157.211i	0.178446i	−0.996012	+	0.0892229i	$0.971562\pi$
$882$	0	0
$883$	14.1471i	0.0160216i	0.999968	+	0.00801079i	$0.00254994\pi$
$883$	14.1471i	0.0160216i	−0.999968	+	0.00801079i	$0.997450\pi$
$884$	115.124i	0.130230i
$885$	0	0
$886$	53.9250	0.0608635
$887$	1741.26	1.96309	0.981547	−	0.191220i	$-0.0612444\pi$
$887$	1741.26	1.96309	0.981547	+	0.191220i	$0.0612444\pi$
$888$	0	0
$889$	215.110	0.241969
$890$	0	0
$891$	0	0
$892$	− 167.737i	− 0.188046i
$893$	−646.575	−0.724048
$894$	0	0
$895$	0	0
$896$	29.9333i	0.0334077i
$897$	0	0
$898$	2.16054i	0.00240595i
$899$	1971.06i	2.19251i
$900$	0	0
$901$	2096.37	2.32671
$902$	1310.79	1.45321
$903$	0	0
$904$	−530.047	−0.586336
$905$	0	0
$906$	0	0
$907$	− 1290.67i	− 1.42301i	−0.702681	−	0.711505i	$-0.748014\pi$
$907$	− 1290.67i	− 1.42301i	0.702681	−	0.711505i	$-0.251986\pi$
$908$	189.220	0.208392
$909$	0	0
$910$	0	0
$911$	906.341i	0.994886i	0.867497	+	0.497443i	$0.165728\pi$
$911$	906.341i	0.994886i	−0.867497	+	0.497443i	$0.834272\pi$
$912$	0	0
$913$	− 1408.63i	− 1.54286i
$914$	− 199.427i	− 0.218191i
$915$	0	0
$916$	637.559	0.696025
$917$	24.4183	0.0266285
$918$	0	0
$919$	−1738.74	−1.89199	−0.945994	−	0.324185i	$-0.894910\pi$
$919$	−1738.74	−1.89199	−0.945994	+	0.324185i	$0.894910\pi$
$920$	0	0
$921$	0	0
$922$	14.7733i	0.0160231i
$923$	72.5076	0.0785564
$924$	0	0
$925$	0	0
$926$	− 225.999i	− 0.244059i
$927$	0	0
$928$	229.601i	0.247415i
$929$	− 529.634i	− 0.570112i	−0.958511	−	0.285056i	$-0.907988\pi$
$929$	− 529.634i	− 0.570112i	0.958511	−	0.285056i	$-0.0920122\pi$
$930$	0	0
$931$	69.2683	0.0744021
$932$	375.664	0.403073
$933$	0	0
$934$	82.1324	0.0879362
$935$	0	0
$936$	0	0
$937$	− 1354.47i	− 1.44553i	−0.691091	−	0.722767i	$-0.742870\pi$
$937$	− 1354.47i	− 1.44553i	0.691091	−	0.722767i	$-0.257130\pi$
$938$	−270.082	−0.287934
$939$	0	0
$940$	0	0
$941$	1406.13i	1.49429i	0.664662	+	0.747144i	$0.268575\pi$
$941$	1406.13i	1.49429i	−0.664662	+	0.747144i	$0.731425\pi$
$942$	0	0
$943$	− 629.823i	− 0.667893i
$944$	446.567i	0.473059i
$945$	0	0
$946$	−1013.82	−1.07169
$947$	1168.66	1.23406	0.617032	−	0.786938i	$-0.288335\pi$
$947$	1168.66	1.23406	0.617032	+	0.786938i	$0.288335\pi$
$948$	0	0
$949$	113.609	0.119714
$950$	0	0
$951$	0	0
$952$	− 241.508i	− 0.253685i
$953$	65.5411	0.0687735	0.0343867	−	0.999409i	$-0.489052\pi$
$953$	65.5411	0.0687735	0.0343867	+	0.999409i	$0.489052\pi$
$954$	0	0
$955$	0	0
$956$	− 722.179i	− 0.755418i
$957$	0	0
$958$	− 189.606i	− 0.197918i
$959$	73.5971i	0.0767435i
$960$	0	0
$961$	1397.32	1.45403
$962$	158.652	0.164919
$963$	0	0
$964$	−29.1798	−0.0302695
$965$	0	0
$966$	0	0
$967$	− 1095.49i	− 1.13287i	−0.824106	−	0.566436i	$-0.808322\pi$
$967$	− 1095.49i	− 1.13287i	0.824106	−	0.566436i	$-0.191678\pi$
$968$	−220.091	−0.227367
$969$	0	0
$970$	0	0
$971$	143.575i	0.147863i	0.997263	+	0.0739313i	$0.0235546\pi$
$971$	143.575i	0.147863i	−0.997263	+	0.0739313i	$0.976445\pi$
$972$	0	0
$973$	267.415i	0.274836i
$974$	392.952i	0.403441i
$975$	0	0
$976$	10.7642	0.0110289
$977$	−1259.60	−1.28925	−0.644627	−	0.764498i	$-0.722987\pi$
$977$	−1259.60	−1.28925	−0.644627	+	0.764498i	$0.722987\pi$
$978$	0	0
$979$	−1248.85	−1.27564
$980$	0	0
$981$	0	0
$982$	728.285i	0.741634i
$983$	275.176	0.279935	0.139968	−	0.990156i	$-0.455300\pi$
$983$	275.176	0.279935	0.139968	+	0.990156i	$0.455300\pi$
$984$	0	0
$985$	0	0
$986$	− 1852.47i	− 1.87878i
$987$	0	0
$988$	35.2991i	0.0357278i
$989$	487.131i	0.492549i
$990$	0	0
$991$	−1111.90	−1.12200	−0.560998	−	0.827817i	$-0.689583\pi$
$991$	−1111.90	−1.12200	−0.560998	+	0.827817i	$0.689583\pi$
$992$	274.711	0.276927
$993$	0	0
$994$	−152.108	−0.153026
$995$	0	0
$996$	0	0
$997$	− 1008.42i	− 1.01145i	−0.862693	−	0.505727i	$-0.831224\pi$
$997$	− 1008.42i	− 1.01145i	0.862693	−	0.505727i	$-0.168776\pi$
$998$	203.720	0.204129
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3150.3.c.f.449.13		16
3.2	odd	2	inner	3150.3.c.f.449.8		16
5.2	odd	4		3150.3.e.f.701.5		8
5.3	odd	4		630.3.e.b.71.4	✓	8
5.4	even	2	inner	3150.3.c.f.449.1		16
15.2	even	4		3150.3.e.f.701.2		8
15.8	even	4		630.3.e.b.71.6	yes	8
15.14	odd	2	inner	3150.3.c.f.449.12		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
630.3.e.b.71.4	✓	8	5.3	odd	4
630.3.e.b.71.6	yes	8	15.8	even	4
3150.3.c.f.449.1		16	5.4	even	2	inner
3150.3.c.f.449.8		16	3.2	odd	2	inner
3150.3.c.f.449.12		16	15.14	odd	2	inner
3150.3.c.f.449.13		16	1.1	even	1	trivial
3150.3.e.f.701.2		8	15.2	even	4
3150.3.e.f.701.5		8	5.2	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	3150.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.41421 q^{2} +2.00000 q^{4} +2.64575i q^{7} +2.82843 q^{8} +O(q^{10})\)	\(q+1.41421 q^{2} +2.00000 q^{4} +2.64575i q^{7} +2.82843 q^{8} -14.1001i q^{11} -1.78360i q^{13} +3.74166i q^{14} +4.00000 q^{16} -32.2729 q^{17} -9.89548 q^{19} -19.9406i q^{22} -9.58126 q^{23} -2.52238i q^{26} +5.29150i q^{28} +40.5881i q^{29} +48.5626 q^{31} +5.65685 q^{32} -45.6408 q^{34} +62.8976i q^{37} -13.9943 q^{38} +65.7349i q^{41} -50.8420i q^{43} -28.2003i q^{44} -13.5500 q^{46} +65.3405 q^{47} -7.00000 q^{49} -3.56719i q^{52} -64.9575 q^{53} +7.48331i q^{56} +57.4003i q^{58} +111.642i q^{59} +2.69105 q^{61} +68.6778 q^{62} +8.00000 q^{64} +72.1824i q^{67} -64.5458 q^{68} +40.6525i q^{71} +63.6965i q^{73} +88.9506i q^{74} -19.7910 q^{76} +37.3055 q^{77} +82.3035 q^{79} +92.9631i q^{82} +99.9021 q^{83} -71.9014i q^{86} -39.8812i q^{88} -88.5703i q^{89} +4.71895 q^{91} -19.1625 q^{92} +92.4054 q^{94} +117.848i q^{97} -9.89949 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 32 q^{4}+O(q^{10}) \) 16 * q + 32 * q^4	\( 16 q + 32 q^{4} + 64 q^{16} + 160 q^{19} + 224 q^{31} - 192 q^{34} - 64 q^{46} - 112 q^{49} - 288 q^{61} + 128 q^{64} + 320 q^{76} + 128 q^{79} - 448 q^{91}+O(q^{100}) \) 16 * q + 32 * q^4 + 64 * q^16 + 160 * q^19 + 224 * q^31 - 192 * q^34 - 64 * q^46 - 112 * q^49 - 288 * q^61 + 128 * q^64 + 320 * q^76 + 128 * q^79 - 448 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more